11521

10584

كارشناسي ارشد

رياضي محض(جبر)

اصفهان: دانشگاه صنعتي اصفهان، دانشكده علوم رياضي

1395

هشت، 77ص.

به فارسي و انگليسي

1395/07/18

رياضي

ID10584

Small and Large Ideals of an Associative Ring PEYMAN SHANBEDI p shanbedi@math iut ac ir June 18 2016 Master of Science Thesis in Farsi Departement of Mathematical Sciences Isfahan University of Technology Isfahan 84156 8311 IranSupervisor Dr Mahmood Behboodi mbehbood@cc iut ac irAdvisor Dr Atefeh Ghorbani a ghorbani@cc iut ac ir2000 MSC Primary 16D25 Secondary 13A15Keywords Artinian ring ascending chain condition cardinality cofinality discrete valuation ring divi sion ring generalized continuum hypothesis Noetherian ring Abstract This M Sc thesis is based on the following paperGreg Oman Small and Large Ideals of an Associative Ring J Algebra Appl 13 2014 no 5 1350151 20 pp Let R be an associative ring with identity and let I be an left right two sided ideal of R Say that I is small if I R and large if R I R In this Thesis we present results on small and large ideals In particular we study their interdependence and how they influence the structure of R Conversely we investigate how the idealstructure of R determines the existence of small and large ideals Rings R for which R I is finite for every nonzero two sided ideal I of R were studied some time ago by Chew andLawn 1 They call such rings residually finite Many of their results were extended in particular to rings withoutidentity by Levitz and Mott in 2 This notion was generalized in the commutative setting by Oman and Salminen To wit let R be a commutative ring and let M be an infinite unitary R module Say that M is homomorphicallysmaller HS if and only if M N M for all nonzero submodules N of M Various structural results on HSmodules were obtained in 3 Dually infinite modules M over a commutative ring for which N M for allproper submodules N of M have also received attention in the literature Gilmer and Heinzer initiated their study calling such modules as J onsson modules Some main results of this thesis are the following Proposition 1 Let R be a ring Then R possesses a left ideal I which is both small and large ifand only if R is finite and not a field We now address two fundamental questions 1 Does the existence of a proper large left ideal imply the existence of a nonzero small left ideal 2 What is the status of the converse The ring Z of integers shows that the answer to the first question is no On the other hand the existence of a nonzerosmall left ideal does imply the existence of a proper large left ideal

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عاطفه قرباني

بيزن طائري، محمدرضا ودادي